Talk:Tricolorability

Reidemeister III figure wrong?

It seems that the "Reidemeister III Move is 3-Colorable" figure should be replaced by a more appropriate one. The 'before' and 'after' in the R-3 move should have the endpoints colored the same color (this is how one proves invariance under 3 colorability). Simply showing two unrelated 3-colorings for the 'before' and 'after' does not seem helpful -- it may even confuse some. Adammanifold (talk) 21:29, 14 May 2009 (UTC)[reply]

The above comment is correct (including the statement about confusion). In fact, it forces one the-same-color junction. --90.180.192.165 (talk) 21:45, 26 October 2009 (UTC)[reply]

Since others agree that this figure is wrong/misleading, I went ahead and created & uploaded a new one. I also cleaned up this comments page so it is more readable overall. Adammanifold (talk) 02:18, 9 December 2009 (UTC)[reply]

In fact, one needs even more than this. We need to see that for *every* tricoloring of the initial state of a Reidemeister 3 move, there is a tricoloring of the final state. 125.63.54.160 (talk) 00:40, 27 January 2014 (UTC)[reply]

Actually, this entire exercise with Reidemeister moves is superfluous in proving the invariance of tricolorability (which is nothing more or less than the existence of a homomorphism of the knot group onto the symmetric group on three letters, and thus invariant by definition). --Ken Perko [lbrtpl@gmail.com] — Preceding unsigned comment added by 24.44.63.248 (talk) 00:48, 19 October 2021 (UTC)[reply]

R-2 Move

(This duplicates HIDDEN comment above) Reidemaister Move II picture is not correct as well. Correct picture should have two parallel (uncrossed) strings at left ("before") picture, cf. Reidemeister_moves. --90.180.192.165 (talk) 21:45, 26 October 2009 (UTC)[reply]

You're right -- the figure that was there was actually the Reidemeister move done twice! I created & uploaded a correct figure. Adammanifold (talk) 02:38, 9 December 2009 (UTC)[reply]

Another comment(s)

>>Reidemeister Move I/II/III does not change tricolorability<< would be more precise than >>Reidemeister Move I/II/III is tricolorable<<. Moreover, a note might be added that >>Some other color combinations (e.g., all-the-same colors) should be considered for complete proof<< --90.180.192.165 (talk) 21:45, 26 October 2009 (UTC)[reply]

I agree that this should be worded better. I'll let you or someone else make these changes. Adammanifold (talk) 02:45, 9 December 2009 (UTC)[reply]

And a final comment

Not only colorability, but hopefully also the number of possible colorings is a knot invariant. I think I have seen it somewhere around here, and I will need it for links just today... --90.180.192.165 (talk) 21:45, 26 October 2009 (UTC)[reply]

This is obvious. The really big payoff from tricolorablity is all the corresponding branched covering spaces (like the one in Heegaard's 1899 thesis that kick-started modern knot theory). As Montesinos and others proved in the 1970s, they include all closed, orientable 3-manifolds. — Preceding unsigned comment added by Ken Perko 24.44.63.97 (talk) 15:00, 6 October 2018 (UTC)[reply]

Tricolorability is a special case of Fox n-colorability (see Fox n-coloring). This notion of counting the different number of possible colorings is covered there, so it probably doesn't need to be covered in this article too; at most it should be very briefly mentioned. Adammanifold (talk) 02:48, 9 December 2009 (UTC)[reply]

Properties section: separable component of a link

The properties section contained the following sentence:

Any link with a separable component is also tricolorable.

I read this as saying that all separable links are tricolorable, which is not true (the crossingless diagram for the 2-component unlink cannot be colored using all three colors). [Actually, it is true -- says I, Ken Perko, an expert on 3-colored covering spaces (see below)] I changed this to the following statement, because I can't think of what else the original author was trying to convey (the added words are in bold).

Any separable link with a tricolorable separable component is also tricolorable.

The above is true, but I feel that the first occurrence of the word separable might be superfluous. If anyone else feels the same way, feel free to remove it. Adammanifold (talk) 17:11, 28 June 2010 (UTC)[reply]

The word "separable" is not superfluous, but the words "with a tricolorable separable component" are [says I, Ken Perko]. After separating a link into two different parts they each can be colored different colors. This establishes 3-colorability (a homomorphism of the link group onto the symmetric group on three letters) even though only two colors are used.

How to color a knot citation

The how to is an algorithm that is incomplete as it stands and there is doubt a greedy algorithm like this would always work without backtrack. Is there a citation or was this just made up? Dmcq (talk) 16:38, 17 July 2010 (UTC)[reply]

I've remove it with this diff as I don't think there is any such theorem. Dmcq (talk) 12:44, 20 July 2010 (UTC)[reply]

Assessment comment

The comment(s) below were originally left at Talk:Tricolorability/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Pretty, but needs some expansion. Geometry guy 23:09, 9 June 2007 (UTC) The picture showing the tricolorability of the Reidemeister move II is incorrect. It should show two strands becoming two strangs overlapped. The picture shows two steps of RMII at once. —Preceding unsigned comment added by 71.158.128.254 (talk) 18:42, 27 September 2007 (UTC)[reply]

Last edited at 18:43, 27 September 2007 (UTC). Substituted at 02:39, 5 May 2016 (UTC)

Comments moved from the article page

I saw some content which looks more like a comment than actually something that should appear in the article page, so I move them here:

Actually, tricolorability is binary only for knots. For links there are other possibilities (which I call "partial tricolorability" -- where some components are left uncolored). See, for example, the Borromean Rings partial 3-coloring at the end of my 2014 paper "Remarks on the History of the Classification of Knots." --Ken Perko (In fact, it's only binary for knots with fewer than three bridges, if one takes account of the number of different 3-colorings modulo mere renumbering of the colors.)

Z423x5c6 (talk) 13:35, 24 January 2022 (UTC)[reply]

Incorrect proof

The current proof uses a structural induction with Reidemeister moves. This is a neat little proof, but structural induction like this only works for knots with finite crossings while the theorem holds even for wild knots. This can be proven by showing that a tricoloring implies the existence of a nontrivial homomorphism from the fundamental group of ambient space to the dihedral group of order 6. I don't have a reference for this proof though. I think ideally the article should cover both proofs (or perhaps a different proof on the infinite crossing case) but mention that the structural induction proof doesn't work in general. However more pressing is that fact that the whole section is unreferenced. I'm editting it with the word tame and tagging it as needing a reference. AquitaneHungerForce (talk) 19:46, 7 July 2024 (UTC)[reply]